If f(x) is strictly increasing function then h(x) is non monotonic function given

Options

A. ` a in (0,3)`
B. `a in (-2,2)`
C. `(3,oo)`
D. `a in (-oo,0)cup (3,oo)`

Explanation

To solve the problem, we need to analyze the function \( h(x) \) based on the given strictly increasing function \( f(x) \). Let's break down the steps to find the intervals for \( a \) such that \( h(x) \) is non-monotonic.

### Step 1: Understand the given function
We know that \( f(x) \) is a strictly increasing function. This means that for any two values \( x_1 < x_2 \), we have \( f(x_1) < f(x_2) \).

### Step 2: Define the function \( h(x) \)
The function \( h(x) \) is given in terms of \( f(x) \). From the video transcript, it seems that \( h(x) \) can be expressed as:
\[ h(x) = f(x)^2 - 2f(x) + 1 \]
This can be rewritten as:
\[ h(x) = (f(x) - 1)^2 \]
This indicates that \( h(x) \) is a square of a function, which is always non-negative.

### Step 3: Analyze the monotonicity of \( h(x) \)
To determine the monotonicity of \( h(x) \), we need to find the derivative \( h'(x) \):
\[ h'(x) = 2(f(x) - 1)f'(x) \]
Since \( f(x) \) is strictly increasing, \( f'(x) > 0 \). The sign of \( h'(x) \) will depend on the term \( (f(x) - 1) \):
- If \( f(x) > 1 \), then \( h'(x) > 0 \) (h is increasing).
- If \( f(x) < 1 \), then \( h'(x) < 0 \) (h is decreasing).
- If \( f(x) = 1 \), then \( h'(x) = 0 \) (h has a critical point).

### Step 4: Find the critical points
To find where \( h(x) \) is non-monotonic, we need to find the values of \( x \) where \( f(x) = 1 \). This requires solving the equation:
\[ f(x) = 1 \]
Assuming \( f(x) \) is a quadratic function, we can express it as:
\[ f(x) = ax^2 + bx + c \]
We need to find the roots of the equation:
\[ ax^2 + bx + (c - 1) = 0 \]
Using the quadratic formula, the roots are given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4a(c - 1)}}{2a} \]
The discriminant \( D = b^2 - 4a(c - 1) \) must be greater than or equal to zero for real roots.

### Step 5: Determine the intervals for \( a \)
From the video, it appears that we need to analyze the expression \( 4a^2 - 12a \) to find the values of \( a \) for which \( h(x) \) is non-monotonic. Setting the discriminant greater than zero:
\[ 4a^2 - 12a > 0 \]
Factoring gives:
\[ 4a(a - 3) > 0 \]
This inequality holds true when:
1. \( a < 0 \)
2. \( a > 3 \)

### Conclusion
Thus, the values of \( a \) for which \( h(x) \) is non-monotonic are:
\[ a \in (-\infty, 0) \cup (3, \infty) \]
And \( h(x) \) is monotonic in the interval \( (0, 3) \).